Figure 1: Smale Horseshoe. (This image was produced by Bill Casselman, Graphics Editor of the Notices of the American Mathematical Society. It first appeared in an article by M. Shub in the May, 2005 issue of the Notices of the AMS.)

The Smale horseshoe is the hallmark of chaos. With striking
geometric and analytic clarity it robustly describes the homoclinic
dynamics encountered by Poincaré and studied by Birkhoff,
Cartwright-Littlewood and Levinson. We give the example first and
the definitions later.

Consider the embedding \(f\) of the disc \(\Delta \) into itself
exhibited in the figure. It contracts the semi-discs \(A\ ,\) \(E\) to the
semi-discs \(f(A)\ ,\) \(f(E)\) in \(A\ ;\) and it sends the rectangles \(B\ ,\)
\(D\) linearly to the rectangles \(f(B)\ ,\) \(f(D)\ ,\) stretching them
vertically and shrinking them horizontally. In the case of \(D\ ,\) it
also rotates by 180 degrees. We don't really care what the image
\(f(C)\) of \(C\) is as long as it does not intersect the rectangle \(B\cup C \cup D\ .\) In the figure it is placed so that the total image
resembles a horseshoe, hence the name.

It's easy to see that \(f\) extends
to a diffeomorphism of the
\(2\)-sphere to itself. We
also refer to the extension
as \(f\ ,\) and work out its
dynamics in \(\Delta \ ,\) i.e., its
iterates
\(f^n\) for \(n \in \mathbb{Z}\ .\)

Necessarily there are three fixed points \(p, q, s.\) The point
\(q\) is a sink in the sense that all points \(z \in A \cup E \cup C\)
converge to \(q\) under forward iteration, \(f^n(z) \rightarrow q\) as
\(n \rightarrow \infty\ .\)

The points
\(p\ ,\) \(s\) are saddle points. If
\(x\) lies on the horizontal through
\(p\) then \(f^n\) squeezes it to
\(p\) as
\(n \rightarrow \infty\ ,\) while if
\(y\) lies on the vertical through
\(p\) then the inverse
iterates of \(f\) squeeze it to
\(p\ .\) With respect to linear
coordinates centered at \(p\ ,\)
\(f(x, y) = (kx, my)\) where \((x,
y) \in B\) and \(0 < k < 1 < m\ .\)
Similarly, \(f(x, y) = (-kx, -my)\) with respect to linear
coordinates on \(D\) at \(s\ .\)

The sets
\[
W^s = \{z : f^n(z)
\rightarrow p\] as \(n \rightarrow +\infty \}\)
\[
W^u = \{z : f^n(z)
\rightarrow p\] as \(n \rightarrow -\infty \}\)
are the stable and unstable
manifolds of \(p\ .\) They
intersect at \(r\ ,\) which is what
Poincare called a homoclinic point.
The homoclinic point here is transverse in the sense that the stable and unstable
manifolds are not tangent at \(r\ .\)
The figure only shows these
invariant manifolds locally.
Iteration extends them
globally.

The key part of the dynamics
of \(f\) happens on the
horseshoe
\[
\Lambda = \{ z : f^n(z) \in B
\cup D\] for all \(n \in \mathbb{Z}\}.\)
Everything there is explained as the "full shift on the space of two symbols," (see symbolic dynamics). Take two symbols, \(0\) and \(1\ ,\) and look at the set \(\Sigma \) of all bi-infinite sequences \(a = (a_n)\) where \(n \in \mathbb{Z}\) and for each \(n\ ,\) \(a_n\) is \(0\) or \(a_n\) is \(1\ .\) Thus
\(\Sigma = \{ 0, 1\}^{\mathbb{Z}}\) is homeomorphic to the Cantor
set. The map \(\sigma : \Sigma \rightarrow \Sigma \) that sends \(a =
(a_n)\) to \(\sigma (a) = (a_{n+1})\) is a homeomorphism called the
shift map. It shifts the decimal point one slot rightward. Every
dynamical property of the shift map is possessed equally by
\(f|_{\Lambda }\) because there is a
homeomorphism
\(h : \Sigma \rightarrow
\Lambda \) such that the
diagram

\(\sigma \) has \(2^n\) periodic orbits of period \(n\ ,\) and so
must \(f|_{\Lambda }\ .\) The set
of periodic orbits of \(\sigma \)
is dense in \(\Sigma \ ,\) and so
must be the set of periodic
orbits
of
\(f|_{\Lambda }\ .\) Small
changes of initial conditions in
\(\Sigma \) can produce large
changes of a \(\sigma \)-orbit,
so the same must be true of
\(f|_{\Lambda }\ .\) In short,
due to conjugacy, the
chaos of
\(\sigma
\) is reproduced exactly in the
horseshoe.

The utility of Smale's analysis is this: every dynamical
system having a transverse homoclinic point, such as \(r\ ,\) is such that some power \(f^T\) has
also a horseshoe containing \(r\ ,\) and has thus the shift chaos.
Nowadays, this fact is not hard to see, even in higher dimensions.
The mere existence of a transverse intersection between the stable
and unstable manifolds of a periodic orbit implies a horseshoe.
In the case of flows, the corresponding assertion holds for the
Poincare map. To recapitulate,

Since transversality persists
under perturbation, it follows
that so does the horseshoe, and
so does its chaos.

The analytical feature of the horseshoe is hyperbolicity
– the squeeze/stretch phenomenon expressed via the derivative.
The derivative of \(f\) stretches tangent vectors which are parallel
to the vertical and contracts vectors parallel to the horizontal,
not only at the saddle points, but uniformly throughout \(\Lambda
\ .\) In general, hyperbolicity of a compact invariant set such as
\(\Lambda\) in any dimension is expressed in terms of expansion and
contraction of the derivative on sub-bundles of the tangent
bundle. Smale unified such examples as the horseshoe and the
geodesic flow on manifolds of negative curvature defining what is
now called uniformly hyperbolic dynamical systems. These are systems in which the non-wandering set
is a uniformly hyperbolic set and its stable and unstable manifolds are transverse at all points or at least exhibit no cycles (see, e.g., the book of Shub below for precise definitions). The study of
these systems has led to many fruitful discoveries in modern
dynamical systems theory.

David Ruelle has
called Smale's 1967 article
"a masterpiece of mathematical
literature." It is still worth reading
today. Hyperbolic dynamics
flourished in the 1960s and 70s.
Anosov
proved the structural stability and ergodicity of the globally
hyperbolic systems that now
bear his name. Sinai initiated the
more general investigation of
the ergodic theory of hyperbolic
dynamical systems, and in
particular showed that the
Markov partitions
of Adler and Weiss could be
constructed for all hyperbolic
invariant sets thus giving a
coding similar the two shift
coding
for the horseshoe. This work
was carried forward by Ruelle
and
Bowen. The invariant measures
they found, now called Sinai-Ruelle-Bowen measures (SRB measures), describe the asymptotic
dynamics of most Lebesgue
points in the manifold even for
dissipative sytems.
Uniformly hyperbolic
dynamical systems are
remarkable.
They exhibit chaotic behaviour.
By the
work of Anosov, Smale, Palis
and Robbin they are structurally
stable or non-wandering set stable, that is the dynamics of a
perturbation of a uniformly
hyperbolic system is
topologically conjugate to the
original globally or at least restricted to the non-wandering sets. By
the work of Sinai, Ruelle, Bowen
they are described statistically.

In the early days of the 60s it
was hoped that uniformly
hyperbolic
dynamical systems might be in
some sense typical. While they
form a large open set on all
manifolds they are not dense.
The search for
the typical dynamical systems
continues to be a great problem.
For progress
see the survey by Bonatti et al. (2004).
Hyperbolic periodic points, their
global stable and unstable
manifolds and homoclinic points
remain some of the principal
features of and tools for
understanding the dynamics of
chaotic
systems.

Indeed transverse homoclinic
points are proven to exist in many
of the dynamical systems
encountered in science and
engineering
from celestial mechanics where
Poincare first observed them
to
ecology and beyond.

History

The history of the discovery of the horseshoe and the state of mathematics in 1960 is described in detail by Smale (1998).

The horseshoe is a natural consequence of a geometrical way of looking at the equations of Cartwright-Littlewood and Levinson. It helps understand the mechanism of chaos, and explain the widespread unpredictability in dynamics. It was discovered in 1960 in Rio de Janeiro, while Dr. Smale was receiving support from the National Science Foundation (NSF) of the United States as a postdoctoral fellow. Dr. Smale was hosted at the Instituto da Matematica, Pura e Aplicada (IMPA), funded by the Brazilian government, which provided a pleasant office and working environment. (Subsequently questions were raised about him having used U.S. taxpayer's money for this research done on the beaches of Rio. In fact none other than President Johnson's science adviser, Donald Hornig, wrote on this issue in 1968 in the widely circulated magazine "Science".)

In Rio, Smale was doing research in an area of mathematics which was to become the theory of chaos. At that time, as a topologist, he prided himself on a paper that he had just published in dynamics. He was delighted with a conjecture in that paper which had as a consequence (in modern terminology) "chaos doesn't exist"! This euphoria was soon shattered by a letter received from an M.I.T. mathematician named Norman Levinson. He had coauthored the main graduate text in ordinary differential equations and was a scientist to be taken seriously. Levinson described an earlier result of his which effectively contained a counterexample to Smale’s conjecture. Levinson’s paper in turn was a clarification of extensive work of the pair of British mathematicians Mary Cartwright and J. L. Littlewood done during World War II. Cartwright and Littlewood
had been analyzing some equations that arose in doing war-related studies involving radio waves. They had found unexpected and unusual behaviour of solutions of these equations. In fact Cartwright and Littlewood had proved mathematically that signs of chaos could exist, even in equations that arose naturally in engineering. But the world wasn't ready to listen, and even today their important contributions to chaos theory are not well-known. To understand Levinson’s counter-example, it was necessary to translate his analytic arguments into geometric way of thinking, which lead into the discovery of the horseshoe.